Name: General Term of an AP: Example
Uploaded: 2019-07-19
Duration: 4 min 25 s
Description: Do you know how to find the first five terms of a sequence in A.P with the help of any two given terms of the sequence? Watch this video to learn how.

Question

Find the sum of the integers between $100$ and $200$ which are not divisible by $9$ .

Accepted Answer

The numbers (integers) between $100$ and $200$ which is divisible by $9$ are $108, 117, 126 . . . . . . . . 198$

Let $n$ be the number of terms between $100$ and $200$ which are divisible by $9 .$

Here, $a = 108, d = 117 - 108 = 9, a_{n} = l = 198$

$∵ a_{n} = l = a + (n - 1) d$

$\Rightarrow 198 = 108 + (n - 1) 9$

$\Rightarrow 90 = (n - 1) 9$

$\Rightarrow n - 1 = 10$

$\Rightarrow n = 11$

$∴$ Sum of terms between $100$ and $200$ which is divisible by $9$ ,

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

$\Rightarrow S_{11} = \frac{11}{2} [2 (108) + (11 - 1) 9] = \frac{11}{2} [216 + 90]$

$= \frac{11}{2} \times 306 = 11 \times 153 = 1683$

Now, the sum of the integers between $100$ and $200$ which are not divisible by $9$ $=$

(sum of total numbers between $100$ and $200$ ) $-$ (sum of total numbers between $100$ and $200$ which are divisible by $9$ ).

Here, $a = 101$ , $d = 102 - 101 = 1$ and $a_{n} = l = 199$

$∵ a_{n} = l = a + (n - 1) d$

$\Rightarrow 199 = 101 + (n - 1) 1$

$\Rightarrow (n - 1) = 98 \Rightarrow n = 99$

$∴$ Sum of terms between $100$ and $200$ ,

$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$

$\Rightarrow S_{99} = \frac{99}{2} [2 (101) + (99 - 1) 1] = \frac{99}{2} [202 + 98]$

$= \frac{99}{2} \times 300 = 99 \times 150 = 14850$

Therefore, the sum of the integers between $100$ and $200$ which is not divisible by $9$

$= 14850 - 1683 = 13167$

Hence, the sum of the integers between $100$ and $200$ which are divisible by $9$ is $1683$ and also not divisible by $9$ is $13167$ .

Find the sum of the integers between $100$ and $200$ which are not divisible by $9$ .

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

Find the sum of the integers between 100 and 200 which are not divisible by 9.

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

Find the sum of the integers between $100$ and $200$ which are not divisible by $9$ .